Small ball probabilities and large deviations for grey Brownian motion

We show that the uniform norm of generalized grey Brownian motion over the unit interval has an analytic density, excluding the special case of fractional Brownian motion. Our main result is an asymptotic expansion for the small ball probability of generalized grey Brownian motion, which extends to other norms on path space. The decay rate is not exponential but polynomial, of degree two. For the uniform norm and the H\"older norm, we also prove a large deviations estimate.


Introduction
Generalized grey Brownian motion (ggBm) is a two-parameter stochastic process B α,β , which is in general not Gaussian.Introduced in [16,17], ggBm has been considered in the physics literature to model anomalous diffusions with non-Gaussian marginals, including both slow (variance grows slower than linearly) and fast diffusive behavior.The process B α,β has stationary increments and is self-similar with parameter H = α/2 [16,Proposition 3.2].The marginal density of ggBm satisfies a fractional partial integro-differential equation [16].Special cases of ggBm include fractional Brownian motion (fBm; β = 1), grey Brownian motion ( [22]; α = β), and Brownian motion (α = β = 1).Our focus is mainly on the case β < 1.In [16], a generalized grey noise space is defined, motivated by white noise space, but with the Gaussian characteristic function replaced by the Mittag-Leffler function.The ggBm is then defined by evaluating generalized grey noise at the test function 1 [0,t) .We do not go into details, because for our purposes, the representation which was proved in [17], is more convenient.Here, B α/2 is a fBm with Hurst parameter H = α/2, and L β is an independent positive random variable whose density is the M-Wright function (see below).The representation (1.1) makes sense also in the limiting case β = 1, but we will not require this.The problem of small ball probabilities, also called small deviations, consists of estimating asymptotically.More generally, we can consider where • is a norm on C γ 0 [0, 1], the space of γ-Hölder continuous functions, with 0 < γ < H = α/2.For ggBm with β < 1, our main result (Theorem 2.3) shows that (1.2) is of order ε 2 , and that this also holds for some other norms.For Gaussian processes, such as fBm (β = 1), the small ball problem has been studied extensively [13], and exponential decay is typical.But there are also many works studying small ball probabilities for non-Gaussian processes; see, e.g., [1,2] and the references therein.We refer to [11,18] for other examples of processes with the small ball rate ε 2 of ggBm.
In Section 2, we will show that the known exponential small ball estimates for fBm can be used to deduce our quadratic small ball estimate for ggBm.As a byproduct, we show that the uniform norm (sup norm) of ggBm has a smooth, even analytic, pdf.In Section 3, we provide a large deviations estimate.The decay rate is exponential, but slower than Gaussian, depending on the parameter β.
Notation: When we write B α,β , we always mean the process on the time interval [0, 1], i.e.B α,β = (B α,β (t)) 0≤t≤1 .We write F H for the cdf of B H , assuming that the choice of the norm • is clear from the context.As usual, R + = (0, ∞) denotes the positive reals.The letter C denotes various positive constants.

Analyticity of the cdf and small ball probability
The M-Wright function, which is the pdf of L β in (1.1), is defined by It is not obvious that M β is a pdf; for this, and more information on M β and its generalizations, we refer to [15].For later use, we note that it follows from Euler's reflection formula that (cf. [23, p. 41] and [15, (3.8)]), which shows, by Stirling's formula for the gamma function, that the series in (2.1) defines an entire function.For this, the crude version of Stirling's formula suffices.We will also need the asymptotic behavior of M β at infinity [15, (4.5)], Our main assumption is that fBm satisfies an exponential small ball estimate w.r.t. to the chosen norm • .
For the uniform norm, it is known that this holds with θ = 1/H, Assumption 2.1 also holds for the γ-Hölder norm, where 0 < γ < H, and for the L 2 -norm.See [4,13,14] for the corresponding values of θ, and for much more information on small ball probabilities for fBm and other Gaussian processes.
The examples we just mentioned are norms in the classical sense, and so we stick to this terminology in our statements.From our proofs, it is clear that it would suffice throughout to assume that • is a measurable non-negative homogeneous functional.Proposition 2.2.Let 0 < α < 2, 0 < β < 1.If the norm • satisfies Assumption 2.1, then the cdf of B α,β is an analytic function on R + .In particular, this holds for the cdf of B α,β ∞ = sup 0≤t≤1 |B α,β (t)|.
Proof.Recall that F H denotes the cdf of B H . From (1.1) we find As M β extends to an entire function (see above), the last integrand clearly is an entire function of ε for any fixed y > 0. The function M β is bounded on R + , as follows, e.g., from (2.1) and (2.4).Thus, the integrand in (2.6) can be bounded by an integrable function of y, independently of ε.Thus, the conditions of a standard criterion for complex differentiation under the integral sign [8, Theorem IV.5.8] are satisfied, which yields the assertion.
Note that fBm, i.e. β = 1, is not covered by Proposition 2.2.In [12], it is shown by Malliavin calculus that sup 0≤t≤1 B H (without the absolute value) has a C ∞ density.
We now show that, for β < 1, the small ball ball probability of ggBm is of order ε 2 as ε ↓ 0. For 2/θ + β < 1 (α + β < 1 for the uniform norm), we express it as a power series, which yields a full asymptotic expansion.We write for the negative moments of the norm of fBm, omitting the dependence on the norm • in the notation η k (H).By integration by parts, it is easy to see that η k (H) is finite under Assumption 2.1.
In the preceding proof, we applied the following estimate for negative moments of the supremum of fBm.Note that moments with positive exponent are estimated in [21]; see also [20].Proof.We show only the upper estimate, as the lower one can be proven analogously.By (2.5), there is ε 0 > 0 such that note that the right hand side is ≥ 1 for y ≥ ε 0 .This implies by Stirling's formula (2.3) for the gamma function.
If 2/θ + β < 1, then the series in (2.8) diverges for any ε > 0. Indeed, there is an increasing sequence (n j ) in N such that the lower bound dist(1 − β − βn j , Z) ≥ C > 0, j ∈ N, holds.For rational β ∈ (0, 1), this is clear by periodicity.For irrational β, it follows from the classical fact that the sequence of fractional parts {nβ} is dense in [0, 1] (Kronecker's approximation theorem).Hence, again by (2.2) and Stirling's formula, , which, together with Lemma 2.4, shows divergence.We leave it as an open problem if (2.8) still holds in the sense of an asymptotic expansion of the small ball probability, if 2/θ + β ≤ 1.

Large deviations
For fractional Brownian motion, it is well known that Indeed, the upper estimate follows from and the Borell-TIS inequality [19,Theorem 4.2], and the lower one is clear from sup 0≤t≤1 |B H (t)| ≥ B H (1). The following result gives a large deviation estimate for ggBm.For β = 1, the distribution has a Gaussian upper tail, of course.For 0 < β < 1, the decay is between exponential and Gaussian, which is sometimes called compressed exponential.
Analogously, we can increase the constants in the exponents to find lower estimates, for which the following reasoning is analogous, and yields (3.3).Therefore, we only discuss the upper estimate for h(y) 1 . This is a straightforward application of the Laplace method [5,Chapter 4] to the integral h(y) which results from the two preceding estimates.The integrand is a strictly concave function with a maximum at for some constant c > 0. As we are not concerned with lower order terms, it suffices to evaluate the integrand at x 0 (y) to conclude  This completes the proof.
We now comment on applying Theorem 3.1 to other norms than the sup norm, which requires verifying (3.2).As mentioned above, for the sup norm, this follows from the Borell-TIS inequality.For an arbitrary norm • on Hölder space, we have In principle, this is in the scope of the general LDP (large deviation principle) for Gaussian measures [6, Theorem 3.4.12],but it may not be trivial to verify the assumptions.For H = 1 2 and the Hölder norm, this was done in [3], extending Schilder's theorem.Note that choosing a stronger topology than the uniform one enlarges the dual space of path space, making it harder to verify the defining property of a Gaussian measure.For the Hölder topology, we are on safe grounds, though, by another approach: Using the double sum method for Gaussian fields, Fatalov has shown that (3.1) holds for the γ-Hölder norm [9, Theorem 1.3], and so Theorem 3.1 is applicable to this norm (with 0 < γ < H, of course).
this is of faster decay than (3.4).